Quotients of surface groups and homology of finite covers via quantum representations

I will show how from Witten-Reshetikhin-Turaev TQFT, we can produce
interesting representations of surface groups. The key fact is the
following : these "quantum representations" of surface groups have
infinite images but every simple loop acts with finite order. Using this
key fact and integral TQFT, we will see how to build regular finite covers
of surfaces where the integral homology is not generated by pullbacks of
simple closed curves on the base.
This talk represents joint work with T.Koberda.